Both Sinn Fein and the DUP have rejected a reported last ditch attempt from the EU to find a Brexit solution.
The Times has reported that the EU is willing to agree to a mechanism to allow the Northern Ireland Assembly to leave the backstop after a set period of time, potentially 2025.
The EU are also reportedly considering a mechanism that would require both unionists and nationalists in the assembly to agree on when to end alignment with Brussels post-Brexit.
“A landing zone on consent could be a double majority within Stormont, to leave, not to continue with the arrangements after X years," a European source told the Times.
The EU wants the UK to accept a customs border in the Irish Sea in return for giving both sides of Stormont a say.
“With the consent mechanism there is no real political reason for the UK to have a separate customs border that is distinct from another regulatory border as Northern Ireland can decide,” the European source said.
EU negotiators are believed to see the deal as a potential breakthrough in an attempt to reach a deal before a summit of European leaders in Brussels next week.
DUP MP Sir Jeffrey Donaldson was quick to reject reports of the proposal.
"This creates a customs border between Northern Ireland and our biggest market in Great Britain," he Tweeted.
"We do more business with GB than the rest of world. Economic madness.
"The DUP has been clear in opposition to this. It’s the Northern Ireland only Backstop reheated and represents no real compromise."
Sir Jeffrey's party colleague Sammy Wilson said that the offer had already been ruled out by the UK government and described it as a "non-runner".
"Whilst the United Kingdom government has put forward sensible and realistic proposals, it is disappointing that the European Union has retreated to its previous failed plans," the DUP Brexit spokesperson said.
"This proposal not only rewrites but actually turns the Belfast Agreement’s consent mechanism on its head. Northern Ireland would be trapped until Sinn Fein consented it could leave."
Sinn Fein vice-President Michelle O'Neill also immediately rejected the proposal.
"There can be no situation in which the local assembly is afforded a veto, the EU have been very clear on that point, they have made it consistently," the Mid-Ulster MLA told Sky News.
"They are not going to afford a regional assembly with an opportunity to thwart any progress that potentially could be reached between the British Government and the EU, that's where the level of agreement needs to be reached.
"The assembly can't be afforded any veto and particularly the DUP can't be afforded a veto, because they have actively worked against the best interests of the people who live on the island of Ireland. "
Mrs O'Neill said that no veto would be considered by Sinn Fein.
"Any situation which would even countenance giving the DUP a say is just not acceptable. Not just to Sinn Fein, it has been roundly rejected by the business community and the other pro-Remain parties and rightly so," she said.
"We're close on coming up to 1000 days where we don't have an assembly working, we don't have a government in the North of Ireland. We don't have that for two reasons, one being Brexit, the other being the relationship between the DUP and the Tories.
"There is no conceivable concept whatsoever that could ever convince me or the people who live on the island of Ireland that there would be any good to come from a veto afforded to the Assembly. Lets just rule that out. It's never going to happen."
TUV leader Jim Allister said the proposal was an "arrogant EU aggression that part of the UK must be extracted from the UK’s customs union and single market and aligned with the Republic".
"Following this, those whose whole purpose is to dismember the UK, namely Sinn Fein, should be gifted a veto on any escape from EU/Republic of Ireland captivity," the North Antrim MLA said.
“This, of course, in part is the product of ever contemplating in the first place union-dividing options. The burden on those who started down that road when they endorsed the Prime Minister’s proposals, specifically the DUP, is immense.
“The DUP have played fast and loose with the union and now we see the entirely predictable outcome of that.”